Mapping Focal Branches via Implicit Equations: The Half-Wave Zone Slit Model
[size=85][b] Applet Description: Focal Curves in Near-Field Slit Diffraction[br][/b]This applet derives [b]explicit equations[/b] for the focal curves within the [b]Half-Wave Zone [url=https://www.geogebra.org/m/n62bumu7]Model[/url][/b] of slit diffraction.[b][br]1. Visualization of Stationary Points[br][/b]The ap plet identifies the stationary points of the 2D distribution J(x, y) using a distinct color-coded system:[br][list][*][b]Red & Blue Points:[/b] Represent local [b]maxima[/b] and [b]minima[/b], respectively.[br][/*][*][b]Crosses (Multiple Colors):[/b] Represent [b]saddle points[/b] positioned between the extrema.[br][/*][*][b]Symmetry:[/b] Note that the distribution of these points is symmetric with respect to the y-axis.[/*][/list][b][br]2. From Implicit Equations to Cubic Roots[br][/b]The derivation begins with the implicit equation ([b][url=https://www.geogebra.org/m/qgg4ujte]eq00[/url][/b]) for the focal curves. By eliminating irrational terms, this is transformed into a [b]cubic polynomial equation[/b] ([b][url=https://www.geogebra.org/m/a8h6psbh]eq0[/url][/b]).[br]We solve the general cubic form: a(z)y[sup]3[/sup] + p(z)y[sup]2[/sup] + c(z)y + d(z) = 0[br]where z is treated as a complex variable. This allows us to express the roots in the explicit form y = f(z). All [url=https://www.geogebra.org/material/show/id/v4fvf8nx]symbolic[/url] and numerical computations were performed directly within [b]GeoGebra[/b].[br][b][br]3. Analysis of the Solution Branches[/b][list][*][b]Curve Cu[sub]f1[/sub]:[/b] Corresponds to the real part of the first complex function, f1(z). This branch coincides perfectly with the original implicit function [b]eq00[/b].[/*][*][b]Curve Cu[sub]f2[/sub][/b],[b][sub] [/sub][/b][b]Curve Cu[sub]f3[/sub][/b] [b]:[/b] Correspond to the real part of the complex functions f2(z), f3(z), respectively, and represent subsequent transformations and additional branches of the cubic solution.[br][/*][/list][quote][b][/b][b]Notes:[/b] [br] Rendering complex functions is computationally intensive. Please be patient or use a desktop computer for a smoother experience.[br] Below the applet, you will find images of the CAS transformations and the behavior of the curves related to the solutions f1(z), f2(z), and f3(z).[/quote][/size][quote][/quote]
Obtaining the implicit equation (eq00) of the focal curve:
Obtaining the implicit equation of the focal curve (eq0) from the equation (eq00) by eliminating irrationality from it:
1. The transformation to eliminate irrationality in the focal curve introduces additional branches to the implicit function
[size=85][b]Implicit functions before (a) and after (b) transformation.[/b] As illustrated in the secondary plot, the transformation to eliminate irrationality in the focal curve introduces additional branches to the implicit function.[/size]
2. Branches of the complex solutions for the cubic polynomial
[size=85](a) The implicit function following transformations, reduced to a cubic polynomial form.[br][br](b)–(d) Branches of the complex solutions for the cubic polynomial. As illustrated, the [i]imaginary components[/i] of these complex solutions vanish along the [i]real [/i]axis.[br] While the first branch, [color=#ff0000][b]f1(z)[/b][/color], satisfies the [color=#0000ff][b]original implicit function[/b][/color], the remaining two branches emerge as a result of the subsequent transformations.[/size]